Algebra with Quadratic Commutation Relations for an Axially Perturbed Coulomb–Dirac Field

Karasev, M. V.; Новикова, Е. М.

doi:10.1023/b:tamp.0000049763.86662.16

Cited by 7 publications

(23 citation statements)

References 11 publications

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“…Later on, this method was extended to a wide class of problems with frequency resonances [10], [11]. It has been worked through on a number of physical models [12]- [14].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Pereskokov

2013

Trans. Moscow Math. Soc.

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Abstract. We investigate the second-order Zeeman effect in a magnetic field using irreducible representations of an algebra with the Karasev -Novikova quadratic commutation relations. To each such representation there corresponds a spectral cluster near the energy level of the unperturbed hydrogen atom. Using this model as an example, we describe a general method for constructing asymptotic solutions near the boundaries of spectral clusters based on a new integral representation. We also study the problem of computing quantum averages near the lower boundaries of clusters.

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“…Later on, this method was extended to a wide class of problems with frequency resonances [10], [11]. It has been worked through on a number of physical models [12]- [14].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Pereskokov

2013

Trans. Moscow Math. Soc.

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“…The quantum averaging method. The quantum averaging method is outlined in [2] and [7]; references to the literature are also given there. We here present only the necessary technical formulas.…”

Section: Quantum Averaging and The Resonance Phenomenonmentioning

confidence: 99%

“…The first version is that one of the vectors, H − 3nE or H + 3nE, is zero. This is a degenerate case where no new dynamical algebras different from su (2) can appear; we do not consider this case in detail. The second version is that the fields E and H are orthogonal.…”

Section: Introductionmentioning

confidence: 96%

“…The system is studied using the quantum averaging method and the theory of coherent states for algebras with nonlinear commutation relations, as explained in the introduction in [2]. We show that an algebra with polynomial relations appears in this model under the resonance condition relating the fields E and H. The structure of these relations is the same as in [3], but the irreducible representations used in this case are quite different (not equivalent).…”

Section: Introductionmentioning

confidence: 97%

“…We continue the study performed in [2] and [3] and consider the configuration of the electric and magnetic fields in which the motion of a quantum particle is controlled by dynamical algebras with nonlinear commutation relations that are beyond the scope of the theory of Lie algebras. In this paper, we consider the motion of a particle in a Coulomb field (for example, electron motion in the field of the nucleus of the hydrogen atom) perturbed by homogeneous electric and magnetic fields.…”

Section: Introductionmentioning

confidence: 99%

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Algebra with polynomial commutation relations for the Zeeman-Stark effect in the hydrogen atom

Karasev

Новикова

2005

Theor Math Phys

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We study the Zeeman-Stark effect for the hydrogen atom in crossed homogeneous electric and magnetic fields. A nonhomogeneous perturbing potential can also be present. If the crossed fields satisfy some resonance relation, then the degeneration in the resonance spectral cluster is removed only in the secondorder term of the perturbation theory. The averaged Hamiltonian in this cluster is expressed in terms of generators of some dynamical algebra with polynomial commutation relations; the structure of these relations is determined by a pair of coprime integers contained in the resonance ratio. We construct the irreducible hypergeometric representations of this algebra. The averaged spectral problem in the irreducible representation is reduced to a second-or third-order ordinary differential equation whose solutions are model polynomials. The asymptotic behavior of the solution of the original problem concerning the Zeeman-Stark effect in the resonance cluster is constructed using the coherent states of the dynamical algebra. We also describe the asymptotic behavior of the spectrum in nonresonance clusters, where the degeneration is already removed in the first-order term of the perturbation theory.

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Wave functions of the toda chain with boundary interaction

Iorgov

Shadura

2005

Theor Math Phys

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In this contribution, we give an integral representation of the wave functions of the quantum N -particle Toda chain with boundary interaction. In the case of the Toda chain with one-boundary interaction, we obtain the wave function by an integral transformation from the wave functions of the open Toda chain. The kernel of this transformation is given explicitly in terms of Γ-functions. The wave function of the Toda chain with two-boundary interaction is obtained from the previous wave functions by an integral transformation. In this case, the difference equation for the kernel of the integral transformation admits separation of variables. The separated difference equations coincide with the Baxter equation.

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Algebra with Quadratic Commutation Relations for an Axially Perturbed Coulomb–Dirac Field

Cited by 7 publications

References 11 publications

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Asymptotics of the spectrum of the hydrogen atom in a magnetic field near the lower boundaries of spectral clusters

Algebra with polynomial commutation relations for the Zeeman-Stark effect in the hydrogen atom

Wave functions of the toda chain with boundary interaction

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